Tracking Spectra

Tracking a spectrum means identifying which operator, spectral notion, and supporting evidence occur in a claim. A normalizable eigenvector, an approximate eigenvector, and an eigenvector of a finite matrix can establish different facts. Reading them as interchangeable is a common source of mistakes about bound states and numerical energy levels.

Required background. Spectra of Operators distinguishes a spectral point from an eigenvalue. Helpful background. Projection-Valued Measures supplies the spectral-weight interpretation used below.

Attach the spectrum to a specified operator

Write the operator together with its Hilbert space and domain. A differential expression with different boundary conditions can define different self-adjoint operators and therefore different spectra. A finite-volume box, the full line, and a periodic cell are different spectral problems even when the local differential expression is unchanged.

Next determine the assertion’s level. For self-adjoint $H$, an eigenvalue $\lambda$ has a nonzero vector in $\ker(H-\lambda I)$. A general spectral point need not. A gap is an open interval disjoint from $\sigma(H)$, not merely an interval in which a particular computation found no eigenvectors.

The identity $E_H(\{\lambda\})\ne0$ detects a genuine eigenvalue through its spectral projection. An interval can have nonzero spectral projection even when every singleton in it has zero projection. This is how a sharp observable can have measurement probabilities without normalizable eigenvectors at individual outcomes.

Ask what a small residual actually proves

Let $H$ be self-adjoint, let $\lambda\in\mathbb R$, and let $u\in D(H)$ be normalized. Define its full residual by

$$\varepsilon=\|(H-\lambda I)u\|.$$

Spectral calculus gives

$$\varepsilon^2 =\int_{\mathbb R}(x-\lambda)^2\,d\mu_u(x) \geq \operatorname{dist}(\lambda,\sigma(H))^2,$$

where $\mu_u(\Delta)=\langle u,E_H(\Delta)u\rangle$. Thus $\operatorname{dist}(\lambda,\sigma(H))\leq\varepsilon$. A small full residual locates nearby spectrum, but does not by itself prove that nearby spectrum consists of eigenvalues.

The same calculation controls where this particular state’s spectral weight lies. For $\delta>0$,

$$\|E_H(\mathbb R\setminus(\lambda-\delta,\lambda+\delta))u\|^2 \leq\frac{\varepsilon^2}{\delta^2}.$$

This is a useful reading distinction: localization in energy is already meaningful without identifying a single eigenstate. To infer closeness to a particular eigenspace, one additionally needs a spectral separation statement and a bound that isolates that eigenspace.

An approximate eigenvector need not approach an eigenvector

For multiplication by $x$ on $L^2(-1,1)$, fix $\lambda\in(-1,1)$ and choose a normalized function $u_\eta$ supported inside $(\lambda-\eta,\lambda+\eta)$. Then

$$\|(H-\lambda I)u_\eta\|\leq\eta.$$

Hence $\lambda$ is spectral: arbitrarily accurate approximate eigenvectors rule out a bounded inverse for $H-\lambda I$. Yet an exact eigenvector would have to be supported at the single point $x=\lambda$, a set of Lebesgue measure zero, so it would vanish in $L^2$.

Shrinking the residual has not created a normalizable eigenvector. Any argument that passes from approximate eigenvectors to an actual vector limit needs extra compactness or spectral isolation. Continuous Spectrum without Eigenvectors gives the broader spectral interpretation.

A compressed residual can hide the omitted directions

Suppose a numerical calculation uses the compression $PHP$ on a finite-dimensional trial space, with orthogonal projection $P$. Here the trial space must lie in $D(H)$ so that $H$ acts on each trial vector. A space contained only in the quadratic-form domain can define a variational matrix without making $Hu$ or its Hilbert-space residual meaningful. Under the operator-domain assumption, a normalized trial vector $u=Pu$ may obey

$$PHPu=\lambda u$$

while its full residual is nonzero. The computed equation removes the component $(I-P)Hu$, which is precisely what must be checked to apply the preceding spectral estimate.

For an exact two-dimensional example, take

$$H=\begin{pmatrix}-1&0\\0&1\end{pmatrix}, \qquad u=\frac1{\sqrt2}\begin{pmatrix}1\\1\end{pmatrix}, \qquad P=|u\rangle\langle u|.$$

The compression has eigenvalue zero because $\langle u,Hu\rangle=0$, although $\sigma(H)=\{-1,1\}$. Its compressed residual is zero and its full residual is $\|Hu\|=1$. Even exact arithmetic cannot fix an inference based on the wrong residual.

This example is a warning about a single compression, not a theorem that every convergent discretization fails. Reliable approximation schemes add hypotheses ensuring spectral convergence or supply residual bounds and exclusion estimates.

Keep variational and spectral-exclusion conclusions separate

A normalized trial state gives an upper bound on the spectral bottom of a semibounded Hamiltonian. It does not alone prove a ground state exists. A trial energy below an independently established essential-spectrum threshold can support a bound-state conclusion through the appropriate variational theorem. Both inputs matter.

Likewise, finding one eigenvalue does not exclude others, and failing to find an eigenvalue does not prove a gap. Read claims of absence for their exclusion argument: a coercive estimate, a resolvent bound, a comparison theorem, or another result that controls the whole relevant subspace.

Exercise: read the residual as a probability bound

A normalized domain vector has residual $\varepsilon=0.02$ in specified energy units around a candidate energy $\lambda$. What spectral-weight guarantee follows for the open interval of radius $\delta=0.1$? Does it prove a bound state?

Solution

The weight outside the interval is at most $(0.02/0.1)^2=0.04$, so the weight inside is at least $0.96$. Also, at least one spectral point lies within distance $0.02$ of $\lambda$. Neither conclusion proves that the interval contains an eigenvalue: a continuous spectral distribution can satisfy both. An isolated spectral cluster or a separate bound-state theorem is needed for a stronger conclusion.

References

• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013 — spectral measures and the difference between spectra and eigenvectors.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, Chapters 2–4 — resolvents, spectral types, and variational spectral estimates.